Tag: ear training

Deciding What is Controllable. Also: Transforming the Polyphony of a Gugue

Well Tempered Klavier, Book I, C Major, Prelude

#1

After finishing the piece I simply asked A.B. what he liked and disliked about his rendition.  He mentioned several negative things and then struck on the one thing that I had primarily noticed: that he stopped the harmonic flow of the piece every time he went from the end of one measure to the beginning of the next.   I missed sensing that inexorable connection that energetically pushed me from the chord in one measure to the chord in the next measure.

He said that he had previously tried condensing the piece into a chorale of whole-note chords (each chord took the place of one measure).  However, he had trouble because he couldn’t do it with any speed.   I said that the speed was of the essence of the procedure.  Ideally, each eighth note’s worth of the piece, in terms of its duration, became the duration of one of the chords of the chorale.

Since it was difficult to shift chords that quickly, I recommended to him that he play just the chord a single measure followed by the chord of the next measure, and then stop.   I asked him to play the first chord as if it were a grace note going to the second chord – the latter being held longer.

This he could do.  We repeated the process for each measure going into its next measure.

Now that we had merged two amino acids into a somewhat longer chain of molecules, I asked him to play as written, while I, in the higher treble, waited until he was near the end , but not at the very end, of one measure only then played the chord of that measure as a grace note followed the next measure’s chord.  Only I would get to the second chord before he finished playing the current measure.  That anticipation of my chord connection gave him the necessary push and energy to keep the piece’s harmonic flow going across the bar line.  Then I would remain silent at the second piano until he reached nearly to the end of the new measure at which point I would break in with the chord of the new ‘current’ measure played as a grace note to the chord that governed the measure that was about to start.  And so on.

#2

A.B. remained worried in particular that the pinkie sixteenth note in the right hand at the end of each measure does not connect smoothly to the next sixteenth note in the left hand at the beginning of the next measure.

His default solution was to figure out  exactly how he wants his pinkie to play that note.  I solved his issue by stepping entirely around his approach.  As he played the piece I sang “la la la…”, but starting with 6th 7th and 8th sixteenth notes of one measure and ended with the fifth sixteenth note of the next measure. on notes to the first note.  I then waited too the 6th note of the new measure and began the process of again singing along for 3 + 5 or 8 notes.

In the form I was singing it, with the way I was grouping the notes that I sung versus those I did not sing, I purposely glossing over the connection between the two notes on either side of the bar lines..  It happens automatically.  By spreading a solder, or flux over the end of one measure and the beginning of the next, I effectively made less important the connection of the bar line.

I noticed in my singing that I helped things along by making misplaced crescendo starting on my first note and ending towards the eighth note.  This helped smooth over A.B.’s faulty connection over the bar line.

At this point we moved on to the fugue: transforming the polyphony of a

fugue in C Major, Book One, Well Tempered:

A.B.: why do I find it so difficult to not hold a voice note longer than it is supposed to last, when the note is already meant to continue sounding through a certain number of the next notes in some of the other voices.  For example if the target voice is a quarter note, or longer, and the other voices are enunciating sixteenth notes.

I gave a brief answer: remembering when to lift a sustained note in one voice is the requires the opposite of everything you do right when knowing when to start a note.  It’s the “dark side” of piano technique: it requires doing everything the wrong way; or is it now the right way?

A.B.: why did you do that?  Why was it working?

Joe: I think it is important to have a distinct pre-vision, pre-conception, of what the beginning of the next measure is going to sound like before you get to it.  It is a strange balance of knowing what’s coming and still being surprised by it.

#2

Can we transform the sound the sound of the fugue in the student’s ear?

We experimented using two pianos with re-registering one of the voice of the fugue.  He would choose one voice to play, and transpose either an octave higher or lower than  it was written, while I played the remaining three voices (without the fourth) at the other piano.

Results:  A.B. said:

My voice sounded different than before.  I head it saying and meaning other other things than I had before, but then realizing that it was the same voice with the same names to its notes, just transposed, and that there was at the same time an abiding identity between both versions of the voice, an identity which was preserved, was eternal and fixed  and was impervious to change of octave.   The new stuff that suddenly I heard, in how that voice combined with the other voices, must have been there latent to the note’s names themselves alone, or to say it in another way were just as present as aesthetic and sonic effects when I played that voice in its original octave.

In the future we will have A.B. play just one voice, but in the octave higher than written and the octave lower than written without playing it in the octave is written.  Later again we can transpose one voice by more than one octave.  If it is the soprano voice we can have it sound in the tenor voice’s range or even below the bass voice.  In the case of the bass voice, we can have it sound in the alto voice’s range or so that it is the highest sounding of all the voices.  At any time I can choose to play all four voices and not just three, leaving one voice to him.  Or, he can choose at random to play just the voices, while I play the other two.  Or, three voices.

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When it’s difficult to get from one chord to another

Sorry to have been out of touch for the past two weeks.    I had cataract surgery and was waiting for my eyes to be able to read the computer screen again.  Anyway, I’m fine now, and the hiatus is over.  But please excuse typos and misspellings.

Consider the situation when we try to connect one chord to another chord, but the second chord is a difficult to get to from the first chord, we can do the following.  The solution ultimately lies in not going from one chord to to a second.  We have to break down this apparent cause and effect within time.  Order in time need not dictate to our imagination order in which our body does things.

We let the hand get used to the second chord before playing the first chord.  We play the second a series of times.  After the first time we move the hand just a little bit away from the keyboard and then find the chord again.  Then we can move the hand right (and then left) along the keyboard, horizontally away from the chord, in gradually increasing distances, and each time find your way spontaneously, without thought, without set-up, to the second chord as if you were already on it.  Eventually your hand ‘remembers’ what that chord feels like, and can return to it from any place at all on or off the keyboard; from any position in all three dimensions that the hand can first be removed to, including for instance from your lap.  Of all these infinite places and positions from which the hand might come to return to that chord, just one such possibility is that the hand is first on the chord that is written first.

Memory is like a glue that adheres to a chord like a familiar friend. Benefiting from this fact, we just have to add in a trick with time.  Instead of the ‘first’ chord being followed by the ‘second’ chord, the second chord is there before the first chord.  we must feel that he have already been there, that the glue of the memory causes our hand to automatically be on the notes of the second chord.  I don’t so much mean that because we have practiced the passage, we get ‘used to’ where the second chord lies.  No, this is different.  This is truly being convinced that you are about to do two totally new things, for the very first time, and yet in spite of that, you act like you already know have been where the second is on the keyboard, tactilely, coverage-wise and finger-wise.

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How to Tackle Difficult Pieces, Practiced Simply

A.B.’s lesson on 4/3/19 on the first prelude from Book One of the Well Tempered Klavier

Balancing memory with freshness:

Be surprised and delighted with each new chord (which is to say each new measure).  This is to balance out the impregnation of the piece by memory, from having heard and/or played the piece many times.   Instead create a “beginner’s mind” for whom the new chord is fresh, unexpected, and bathed in morning light.  You just don’t know what’s coming.  Memory doesn’t go away but a proportional balance is attained between memory and the unforeseeableness of the future.

The persistence of a single chord through an entire measure:

In this piece it helps that you were formerly an organist, for as long as you hold the keys down on the organ manual the sounds continue unabated, persistently, and without the piano’s ‘decay’.  Hear in your “inner” ear of imagination the five different notes of each measure as a simultaneous ensemble, which continues unbated as a totality from the beginning of the measure to the end of the measure.

A.B. is not satisfied with his control over the evenness of the sounds in a measure:

Take a single measure out of the flow of the piece.  Reiterate the first note of the measure over and over until it “sounds like you want”.  Do this without thinking of the other notes and whether they will match the first note in sonically – in other words this is not yet about evenness between notes).  Then switch to the second note.  Play it ever and over, until, as before, it sounds how you want.  Repeat this procedure for each further note in the measure.  When you play the measure as written you will notice in retrospect that all the notes were even, although you were in no way trying to match them, but instead having each note have its ‘ideal’ sound.  A musician with a  good ear will always be able to tell when a sound has reached a certain ideal perfection, but not through analysis, through an intuitive sense of the sound.

For evenness when one note, occurring between two other notes, is not balanced sound-wise with the others:

In the measure that begins : f2 f3 a3 c4 e4, the c4 was not balanced with the a3 and e4.  I suggested that he hold down the a3 and e4, and while they are being held, repeat c4 over and over.

Another path to evenness: the written notes are part of a larger whole:

In measure one, for example, turn the measure’s notes into a rapid arpeggio that starts, with the highest pitch, e5, descends through the notes of the chord until reaching the bottom note (c4) and without pause re-ascends to the top note.  This creates a more cohesive and integrated motion in your hand.  Once you have this gestalt, you can remain silent during the first part of this arpeggio and start playing in the middle of it, at the note that is supposed sound first in the measure.  Eventually there is no need to pause or mark time for the first half of the arpeggio, it can occur in the inner feelings of the body in just a split second.

Yet another path to evenness:

When a baton twirler causes the baton to make a circle, it is the result of a sequence of different motions all blended together in a one overall fluid motion.  I’m ignorant of the breakdown of those motions, but you can still imagine, yourself as twirling a baton, one cycle every half measure (as the note pattern repeats).

I would sing a sustained line for A.B.:

Sometimes I would sing a sustained melody, one note per measure, starting at the beginning of each measure, made up of the top note of each measure.  Maybe I thought of doing this because I Gounod’s Ave Maria flitted through my mind.  That Gounod may have felt that the Bach begged for a continuous line (adumbrated by Bach made tangible  by Gounod).  The effect that my singing had unconsciously on A.B. was each note of the measure was instinctively made to balance, or fuse sonically, with the sustained note I was singing.

How to bring out the dramatological curve of a piece, even though it was originally played on an instrument of a constant degree of loudness: 

There are not many overtly dramatic moments in the piece that stand out from the monotonous (sic) patterns that repeat every half measure.

And even if we become aware at a certain time of these moments, they will afterwards fade into the background due to the abrasion or erosion of constant playing of the piece.  So  make the most of these moments.

Here is one example.  Chords outlining diminished chords, for instance, happen only a few times in the piece, but each time it does, try to react to the sound of the chord as being jarring, intense, dissonant.  This effect can be gained even without making any change in the loudness of those measures versus the surrounding measures. One can intimate a dramatic curve merely with intent and adumbration in the flow of the notes.

One of my other students, while playing through the Adagio from  Beethoven’s Op 13, came across of a few measures of diminished chords in the passage leading back to the second A section of its ABA form.  She said “diminished chords are ugly”. I said: that’s great, can you make them sound as ugly as possible!

Another example. When an interval of a minor second in the left hand, treat it as an astonishing, unexpected dissonance.

One more example, this time a longer passage:

In the second half of the page there is a long dominant pedal point in the left hand playing g2 (lowest line of bass clef).  As he went from one measure to the next I repeated: “long … long endeavor … never stops … we’re not ‘there’ yet”.

Matching two sounds that are separated in time:

When you play the first half of a measure and get to the highest note, consciously hold its sound in your ear’s memory, so that when you play the same note in the second half of the measure you can match it with the first.

Sometimes a “group” of notes is just one note:

In the last few measures of the prelude, I find that it is not useful to think of groups of four notes, or even two notes, the measures are too ambiguous compared to what has preceded it throughout the piece. My way around this is to play these last measures in “groups of ONE” note.  To promote this I say out loud as i am playing: “One”, “one”, “one” …. “.  Every note bears little allegiance to every other note except when though of in retrospect.

Remember that your pinkie is part of your hand, not a separate appendage:

Often your pinkie seems to be out in right field, detached from the rest of your hand as if it were a separate appendage.  Hold the pinkie in the unity of your whole hand.

Isolating Variables: the sequence of fingers as against the sequence of pitches:

This is in line with what we just said about the pinkie being “held” in the hand.  In measure three A.B. is using fingers 1, 3 then 5 to play g4 d5 and f5.

I asked him to cover the notes g4-a4-b4-c5-d5 with the five fingers of his right hand.  Play it as a cluster and hold it.  And while holding all five notes try to lift the thumb and replay the G, then again while still holding all the notes, raise the third finger and replay the d5, and similarly with the pinkie for f5.  Just focus on an awareness of the identity of which finger you are playing, as if to say “these are the fingers I’m going to use: 1 3 and 5”.  Then use the same fingers but for the written notes (g4 d5 f5).  You hopefully will feel an interesting transference of the awareness of which fingers to use, now mapped onto a different set of fingers.

Isolating Variables: The sensation of evenness as against any physical actions taken to instill evenness, especially when there is a new set of notes:

There is an ’emotional’, a generalized physical sense in the body as a whole, of ‘balance’ among the notes of the keyboard that are played together and in close succession.  As with any feeling, this emotional state can be reproduced at will under different circumstances.  Rather than the details of how to play the next measure evenly, try to reproduce the experience of having this feeling.

This distinction applies to many situations in playing.

For instance: there is the sensation we get of playing an ascending set of pitches.  This feeling can be conjured up even if we are playing a descending set of pitches.  Sometimes doing this is very useful in a Bach fugue to help homogenize two different voices, so that what a second voice is doing does not sound too dissimilar from what a first voice is doing.

Or, a sense of enlarging and getting louder can overlay a series of notes that are getting softer.

Or, a sense of wide space between the fingers in the hand can overlay a passage that involves a series of notes only one half step apart from each other.

Or, the sense of energy that we get from one very dynamic piece or passage from such a piece, and overlaying that feeling of energy onto all passages, slow or fast, loud or soft.

Making a clear connection between two non-adjacent fingers:

There is a measure in the first part where the pianist plays this sequence of notes: b3 c4 e4 g4 c5 … .

Notice that I tapped your fourth finger when you went from your third finger on g4 to the fifth finger on c5, It was meant to show the hand the focus of the ‘connection’ between the fingers playing g4 and c5, more at being located at the connection between the 3rd and 5th fingers.

At another point in the lesson I slid a pencil between his second and fifth finger.  The pencil passed over those two fingers but passed underneath the fingers in between them.  This helped him sense that those two fingers don’t act separately, but more at being the two ends of the plank of a see-saw, and thus the result of one single action.

More about see-saws:

Regardless of what two fingers play one after the other, and regardless of the distance between the notes they play, always an imaginary see-saw plank between the current note’s finger and the next note’s finger.  Add to this image an almost felt, pivot point, midway between the two fingers.  Now pretend you are a very strong person who  can make the two ends of the plank move reciprocally move up and down just by leaning first on one side and then the other side of where the pivot.

Once you are on the second note resulting from the first see-saw, move the see-saw’s location so that it connects this second note with the note that follows it.

To develop the sense of this see-saw, and the ability to relocate it quickly, it may help (using measure one as an example) to do this exercise:

Go back and forth between c4 and e4 (something which I notate as |: c4 e4 :|.  Once that see saw is functioning organically do the same for |: e4 g4 :|, and so on.

Addendum to the previous section:

It is your tendency, when you encounter a problem in a measure, to  just play ahead for quite a long time, and then tend to the problem later.  It is good to balance that tendency out with the ability to not move ahead, maybe only as far as the end of the current measure, and then focus in on tiny details.  Focusing entails a greater degree of  awareness of what is  happening physical and sound-wise, plus  reiterating that tiny detail until it sounds how you want it to sound.

Don’t rob the last note of each measure of its full duration:

A.B. usually tries to rush into the new hand position at the beginning of the next measure.  He feels that he may not have enough time to do it in, and compensates by holding the last note of the current measure a little shorter than the other notes of the measure.  I said “it is always good to try to hold longer whatever note sounds just before a leap, a skip, or a change of hand position.  One can deal with this near the end of the note by continuing to hold it when your hand tells you it is time to let go of it.  There is another way that is just as effective, that is more at being located time-wise at the beginning of the note rather than near the end.  Start the note with the “intention” of holding it longer.

We reached the goal of evenness:

Joe: in general today we have accomplished one of your goals: the sound is now even throughout.  During the attempt to make each note sound clear and close to its ideal sound, you were finding it easier to do this when playing all the notes a little louder than usual.  Often two variables get tied together, “entangled” as it were.  On the hand playing more evenly, on the other playing more loudly.  The latter helps achieve the former, only at some point, you want to separate the  former from depending on the latter.  Once you have effected this separation, the evenness and clear-speaking-ness of each sound, no longer depends on loudness and can occur at any dynamic you choose.

General comment #1:

 Notice that while you tend to try to solve things with specific actions of specific fingers, I almost never suggest a solution that involves the fingers, but relies instead on a more integrated motion of all the parts of the arm from shoulders to hands.

General comment #2:

I think you are evolving from one species of musician into another species: from an organist to a pianist.

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Microtonal Musings

Microtonal Musings:

1: “in tune” or “out of tune”:

The tonally trained ear expects to hear things in a certain way, and clings to that way in spite of gradually mounting evidence that what they are hearing is not tonal but microtonal.

I’ve demonstrated this with a sound experiment in which a major triad (such as C-E-G) is gradually transformed into a minor triad (C-Eb-G) followed by a reverse direction.  The root note and fifth remain constant; the third is gradually lowered over the course of a certain duration until it has fallen a half step, at which point the third starts rising in pitch, at the same rate as it the pitch was lowered, until it is back to its usual position as the third of a major triad.

In this particular experiment the third is lowered (and later raised) at the rate of one hundredth of a semi-tone (a “cent”) every third of a second.

For many people, no change is noticed for a while.   Their ear continues to hear, or cling to hearing, a major triad – one albeit that is “out of tune” but still clearly intended to be a major triad.  The ear does not accept that it is perceiving a microtonal tonal triad that is neither major nor minor.   The microtonal change is considered an imperfection in the intonation.  There is no recognition of the triad as being of a new aesthetic species – neither major or minor.

Then a sudden switch occurs.   At a certain point in the migration downwards of the third of the triad, most interestingly a point that is closer to the eventual minor chord and further from the initial major chord, the sound, almost instantly, changes in the listener’s ear from being heard as an out of tune major triad to an out of tune minor triad.

This “inaccurate” minor triad persists until the third is close to its final value, at which point the sensation the minor chord at last is getting more and more “in tune”, until at the end it sounds very in tune.

The most interesting part of this sound experiment is that when the third starts traveling in the other direction, the location of the point where the ear ceases to hear the triad as an out of tune minor chord and flips over to hearing it as an out of tune major chord, does not occur at the same point as the similar position during the first phrase of the experiment.  This change in perception now occurs when the triad is closer to the final major chord and further from the minor triad.

2. A self-similar “fractal” chord:

I’ve made many experiments in discovering types of microtonal chords that have a distinct effect that is aesthetically interesting.

My aim was to create a microtonal analogy to a ‘self similar’ fractal design.   The results I got were extremely beautiful, and unlike in case number one, above, could not under any circumstances be heard as an ‘out of tune’ version of a more tonal chord.

I started with an arbitrary selection of a lowest and a highest pitch.  I then inserting a medium pitch that had the effect of dividing the overall range into two parts that bore a certain ratio (in my first experiment this ratio in pitch was 2 : 3).  I continued to divide up each of the smaller pitch intervals by the same ratio.  What started as just two pitches, became successively, hree pitches, five pitches, nine pitches, seventeen pitches, etc..*

*In computing the frequencies of the microtones I started with the unit of the “cent” (one hundredth of a half step) and then converted each cents value to a frequency.

3. Overtone series:

When an ear perceives a single tone or pitch from an orchestral instrument or the voice, an analysis of the sound vibration shows that there are actually a ‘chord’ of different pitches executing separate vibrations.  These additional tones are known as overtones.  If one could separate one overtone from the rest we would hear a sound at a different pitch from the one the ear first perceives.

One of the properties of the overtones is that they are the most spread out in pitch near the ‘fundamental frequency’ (the pitch that the ear perceives) and clump closer and closer together as they continue upwards in pitch.

There is a piece by Stockhausen called “Stimmung”* which has a group of singers each singling one of the upper overtones of a constant fundamental frequency.   A ‘range’ of overtones is chosen by the composer.  The fundamental is never sung, but a consecutive group of overtones is used.  A variable in this selection is what should the lowest pitched overtone in the overtone series that should be sounded and which is the highest pitched one, and how many overtones does that ‘interval’ contains.

In its application to microtonal music, such a group of overtones, can be used as a ‘scale’ of available pitches out of which the notes of the piece are formed.   However, one property of a scale is that it repeats over and over, usually at the octave.  We can make a series of overtones do this by taking just one octave of the overtone series and transposing its pitches up and down various numbers of octaves so as to form a continuous scale from bass to treble.

For most orchestral instruments the overtones are linear in frequency.  The first overtone is twice the fundamental frequency, the second overtone is three times the fundamental frequency, etc..   But the more three dimensional the instrument is the more it deviates from this simple linear pattern.  A bell, for instance, whose vibrating mechanism does not approximate a one dimensional line, has a different arrangement of overtones.

And if a four-dimensional creature were to suspend a four-dimensional bell from a string, and then set it into vibration, there would be an less linear overtone series.  It does not matter that we cannot construct such an instrument, for mathematics enables to predict what the overtones would be, and they can be reproduced exactly on an electronic synthesizer that is set up for microtones.   So we can form scales out of the overtone series for n-dimensional objects (where n goes beyond three).

Here is a list of sample possible constants for generating an ‘altered’ linear overtone series on a particular note:

In the following n is a whole number, and ff the fundamental frequency:

(pi) x (n) x (ff)

(e) x (n) x (ff)

Here are some other possibilities of generating a linear overtone series not based on multiplying the fundamental frequency by whole numbers.

1.1

2 to the 1/2 power

pi times e

sine of an angle

* There was work by Maurice Béjart’s modern ballet company which was set to the music of Stimmung which was sung on stage at Carnegie by the “Swingle Singers”.  Each singer intoned the pitch corresponding to one of the linear overtones of a single fundamental pitch.  So that the tones were not too widely separated pitch-wise, they used a part of the overtone series where there were approximately as many overtones within the scope of one octave as there are notes forming one octave of a more familiar scale.  For instance the following numbered overtones, in the fourth octave about the fundamental, span an octave and divide that octave into 8 parts: 7 8 9 10 11 12 13 14 15  

4. Graphing a mathematical function:

There is an aesthetic fallacy in trying to find a means of translation between something spatial (as a graph) and something temporal (as music).  However, if one is willing to experiment, one could try to derive the notes of microtonal chord from the y-values of some function f(x).  Each next note in the chord would be f(x) for each whole number value of x.  What would a parabola sound like?  A hyperbola?  A sine way?  We don’t know until we ‘hear it’.  We may stumble on a function whose sound as a chord is pleasing and unique aesthetically.

5. Expanding or compressing a tonal piece around a constant center of pitch:

This is more productive of interesting sounding tone groups.  Bach Chorales lend themselves nicely to this procedure.

Take each chord, translate it into cents, and then either increase or decrease each pitch in the chord relative to some stable frequency that is either be one of the pitches of the original chord or a pitch that is chosen randomly but which remains throughout the chorale as the center of expansion and/or contraction.   Or, another way would be to use the notes in one of the four voices as the “stable” pitch (even though it may change from beat to beat) and contract or expand the pitches of the other three voices relative to it.

P.S.

For those of you who dabble in microtones would you let me know what methods you use or whether any of the methods described above have proven useful.  Thanks, Joe

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